Invariant theory for non-associative real two-dimensional algebras and its applications (Q1889939)

The set \({\mathcal A}\) of all non-associative algebra structures on a fixed 2-dimensional real vector space \(A\) is naturally a \(\text{GL}(2,\mathbb R)\)-module. We compute the ring of \(\text{SL}(2,\mathbb R)\)-invariants in the ring of polynomial functions, \({\mathcal P}\), on \({\mathcal A}\). We use invariant theory to compute the exact number of nonzero idempotents of an arbitrary 2-dimensional real division algebra. We show that the absolute invariants (i.e., the \(\text{GL}(2,\mathbb R)\)-invariants in the field of fractions of \({\mathcal P}\)) distinguish the isomorphism classes of 2-dimensional non-associative real division algebras. We show that the (open) set \(\Omega^+\subset{\mathcal A}\) of all division algebra structures on \(A\) has four connected components. A similar result is proved for another class of regular 2-dimensional real algebras (the principal isotopes of the algebra \(\mathbb R\oplus\mathbb R\)).